3.58/1.76	YES
3.58/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.58/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.58/1.77	
3.58/1.77	
3.58/1.77	Termination w.r.t. Q of the given QTRS could be proven:
3.58/1.77	
3.58/1.77	(0) QTRS
3.58/1.77	(1) QTRSToCSRProof [SOUND, 0 ms]
3.58/1.77	(2) CSR
3.58/1.77	(3) CSRRRRProof [EQUIVALENT, 41 ms]
3.58/1.77	(4) CSR
3.58/1.77	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.58/1.77	(6) CSR
3.58/1.77	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.58/1.77	(8) YES
3.58/1.77	
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(0)
3.58/1.77	Obligation:
3.58/1.77	Q restricted rewrite system:
3.58/1.77	The TRS R consists of the following rules:
3.58/1.77	
3.58/1.77	   active(and(true, X)) -> mark(X)
3.58/1.77	   active(and(false, Y)) -> mark(false)
3.58/1.77	   active(if(true, X, Y)) -> mark(X)
3.58/1.77	   active(if(false, X, Y)) -> mark(Y)
3.58/1.77	   active(add(0, X)) -> mark(X)
3.58/1.77	   active(add(s(X), Y)) -> mark(s(add(X, Y)))
3.58/1.77	   active(first(0, X)) -> mark(nil)
3.58/1.77	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
3.58/1.77	   active(from(X)) -> mark(cons(X, from(s(X))))
3.58/1.77	   active(and(X1, X2)) -> and(active(X1), X2)
3.58/1.77	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.58/1.77	   active(add(X1, X2)) -> add(active(X1), X2)
3.58/1.77	   active(first(X1, X2)) -> first(active(X1), X2)
3.58/1.77	   active(first(X1, X2)) -> first(X1, active(X2))
3.58/1.77	   and(mark(X1), X2) -> mark(and(X1, X2))
3.58/1.77	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.58/1.77	   add(mark(X1), X2) -> mark(add(X1, X2))
3.58/1.77	   first(mark(X1), X2) -> mark(first(X1, X2))
3.58/1.77	   first(X1, mark(X2)) -> mark(first(X1, X2))
3.58/1.77	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
3.58/1.77	   proper(true) -> ok(true)
3.58/1.77	   proper(false) -> ok(false)
3.58/1.77	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.58/1.77	   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
3.58/1.77	   proper(0) -> ok(0)
3.58/1.77	   proper(s(X)) -> s(proper(X))
3.58/1.77	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
3.58/1.77	   proper(nil) -> ok(nil)
3.58/1.77	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.58/1.77	   proper(from(X)) -> from(proper(X))
3.58/1.77	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
3.58/1.77	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.58/1.77	   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
3.58/1.77	   s(ok(X)) -> ok(s(X))
3.58/1.77	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
3.58/1.77	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.58/1.77	   from(ok(X)) -> ok(from(X))
3.58/1.77	   top(mark(X)) -> top(proper(X))
3.58/1.77	   top(ok(X)) -> top(active(X))
3.58/1.77	
3.58/1.77	The set Q consists of the following terms:
3.58/1.77	
3.58/1.77	   active(from(x0))
3.58/1.77	   active(and(x0, x1))
3.58/1.77	   active(if(x0, x1, x2))
3.58/1.77	   active(add(x0, x1))
3.58/1.77	   active(first(x0, x1))
3.58/1.77	   and(mark(x0), x1)
3.58/1.77	   if(mark(x0), x1, x2)
3.58/1.77	   add(mark(x0), x1)
3.58/1.77	   first(mark(x0), x1)
3.58/1.77	   first(x0, mark(x1))
3.58/1.77	   proper(and(x0, x1))
3.58/1.77	   proper(true)
3.58/1.77	   proper(false)
3.58/1.77	   proper(if(x0, x1, x2))
3.58/1.77	   proper(add(x0, x1))
3.58/1.77	   proper(0)
3.58/1.77	   proper(s(x0))
3.58/1.77	   proper(first(x0, x1))
3.58/1.77	   proper(nil)
3.58/1.77	   proper(cons(x0, x1))
3.58/1.77	   proper(from(x0))
3.58/1.77	   and(ok(x0), ok(x1))
3.58/1.77	   if(ok(x0), ok(x1), ok(x2))
3.58/1.77	   add(ok(x0), ok(x1))
3.58/1.77	   s(ok(x0))
3.58/1.77	   first(ok(x0), ok(x1))
3.58/1.77	   cons(ok(x0), ok(x1))
3.58/1.77	   from(ok(x0))
3.58/1.77	   top(mark(x0))
3.58/1.77	   top(ok(x0))
3.58/1.77	
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(1) QTRSToCSRProof (SOUND)
3.58/1.77	The following Q TRS is given: Q restricted rewrite system:
3.58/1.77	The TRS R consists of the following rules:
3.58/1.77	
3.58/1.77	   active(and(true, X)) -> mark(X)
3.58/1.77	   active(and(false, Y)) -> mark(false)
3.58/1.77	   active(if(true, X, Y)) -> mark(X)
3.58/1.77	   active(if(false, X, Y)) -> mark(Y)
3.58/1.77	   active(add(0, X)) -> mark(X)
3.58/1.77	   active(add(s(X), Y)) -> mark(s(add(X, Y)))
3.58/1.77	   active(first(0, X)) -> mark(nil)
3.58/1.77	   active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
3.58/1.77	   active(from(X)) -> mark(cons(X, from(s(X))))
3.58/1.77	   active(and(X1, X2)) -> and(active(X1), X2)
3.58/1.77	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.58/1.77	   active(add(X1, X2)) -> add(active(X1), X2)
3.58/1.77	   active(first(X1, X2)) -> first(active(X1), X2)
3.58/1.77	   active(first(X1, X2)) -> first(X1, active(X2))
3.58/1.77	   and(mark(X1), X2) -> mark(and(X1, X2))
3.58/1.77	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.58/1.77	   add(mark(X1), X2) -> mark(add(X1, X2))
3.58/1.77	   first(mark(X1), X2) -> mark(first(X1, X2))
3.58/1.77	   first(X1, mark(X2)) -> mark(first(X1, X2))
3.58/1.77	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
3.58/1.77	   proper(true) -> ok(true)
3.58/1.77	   proper(false) -> ok(false)
3.58/1.77	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.58/1.77	   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
3.58/1.77	   proper(0) -> ok(0)
3.58/1.77	   proper(s(X)) -> s(proper(X))
3.58/1.77	   proper(first(X1, X2)) -> first(proper(X1), proper(X2))
3.58/1.77	   proper(nil) -> ok(nil)
3.58/1.77	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.58/1.77	   proper(from(X)) -> from(proper(X))
3.58/1.77	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
3.58/1.77	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.58/1.77	   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
3.58/1.77	   s(ok(X)) -> ok(s(X))
3.58/1.77	   first(ok(X1), ok(X2)) -> ok(first(X1, X2))
3.58/1.77	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.58/1.77	   from(ok(X)) -> ok(from(X))
3.58/1.77	   top(mark(X)) -> top(proper(X))
3.58/1.77	   top(ok(X)) -> top(active(X))
3.58/1.77	
3.58/1.77	The set Q consists of the following terms:
3.58/1.77	
3.58/1.77	   active(from(x0))
3.58/1.77	   active(and(x0, x1))
3.58/1.77	   active(if(x0, x1, x2))
3.58/1.77	   active(add(x0, x1))
3.58/1.77	   active(first(x0, x1))
3.58/1.77	   and(mark(x0), x1)
3.58/1.77	   if(mark(x0), x1, x2)
3.58/1.77	   add(mark(x0), x1)
3.58/1.77	   first(mark(x0), x1)
3.58/1.77	   first(x0, mark(x1))
3.58/1.77	   proper(and(x0, x1))
3.58/1.77	   proper(true)
3.58/1.77	   proper(false)
3.58/1.77	   proper(if(x0, x1, x2))
3.58/1.77	   proper(add(x0, x1))
3.58/1.77	   proper(0)
3.58/1.77	   proper(s(x0))
3.58/1.77	   proper(first(x0, x1))
3.58/1.77	   proper(nil)
3.58/1.77	   proper(cons(x0, x1))
3.58/1.77	   proper(from(x0))
3.58/1.77	   and(ok(x0), ok(x1))
3.58/1.77	   if(ok(x0), ok(x1), ok(x2))
3.58/1.77	   add(ok(x0), ok(x1))
3.58/1.77	   s(ok(x0))
3.58/1.77	   first(ok(x0), ok(x1))
3.58/1.77	   cons(ok(x0), ok(x1))
3.58/1.77	   from(ok(x0))
3.58/1.77	   top(mark(x0))
3.58/1.77	   top(ok(x0))
3.58/1.77	
3.58/1.77	Special symbols used for the transformation (see [GM04]):
3.58/1.77	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.58/1.77	The replacement map contains the following entries:
3.58/1.77	
3.58/1.77	and: {1}
3.58/1.77	true: empty set
3.58/1.77	false: empty set
3.58/1.77	if: {1}
3.58/1.77	add: {1}
3.58/1.77	0: empty set
3.58/1.77	s: empty set
3.58/1.77	first: {1, 2}
3.58/1.77	nil: empty set
3.58/1.77	cons: empty set
3.58/1.77	from: empty set
3.58/1.77	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(2)
3.58/1.77	Obligation:
3.58/1.77	Context-sensitive rewrite system:
3.58/1.77	The TRS R consists of the following rules:
3.58/1.77	
3.58/1.77	   and(true, X) -> X
3.58/1.77	   and(false, Y) -> false
3.58/1.77	   if(true, X, Y) -> X
3.58/1.77	   if(false, X, Y) -> Y
3.58/1.77	   add(0, X) -> X
3.58/1.77	   add(s(X), Y) -> s(add(X, Y))
3.58/1.77	   first(0, X) -> nil
3.58/1.77	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.58/1.77	   from(X) -> cons(X, from(s(X)))
3.58/1.77	
3.58/1.77	The replacement map contains the following entries:
3.58/1.77	
3.58/1.77	and: {1}
3.58/1.77	true: empty set
3.58/1.77	false: empty set
3.58/1.77	if: {1}
3.58/1.77	add: {1}
3.58/1.77	0: empty set
3.58/1.77	s: empty set
3.58/1.77	first: {1, 2}
3.58/1.77	nil: empty set
3.58/1.77	cons: empty set
3.58/1.77	from: empty set
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(3) CSRRRRProof (EQUIVALENT)
3.58/1.77	The following CSR is given: Context-sensitive rewrite system:
3.58/1.77	The TRS R consists of the following rules:
3.58/1.77	
3.58/1.77	   and(true, X) -> X
3.58/1.77	   and(false, Y) -> false
3.58/1.77	   if(true, X, Y) -> X
3.58/1.77	   if(false, X, Y) -> Y
3.58/1.77	   add(0, X) -> X
3.58/1.77	   add(s(X), Y) -> s(add(X, Y))
3.58/1.77	   first(0, X) -> nil
3.58/1.77	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.58/1.77	   from(X) -> cons(X, from(s(X)))
3.58/1.77	
3.58/1.77	The replacement map contains the following entries:
3.58/1.77	
3.58/1.77	and: {1}
3.58/1.77	true: empty set
3.58/1.77	false: empty set
3.58/1.77	if: {1}
3.58/1.77	add: {1}
3.58/1.77	0: empty set
3.58/1.77	s: empty set
3.58/1.77	first: {1, 2}
3.58/1.77	nil: empty set
3.58/1.77	cons: empty set
3.58/1.77	from: empty set
3.58/1.77	Used ordering:
3.58/1.77	Polynomial interpretation [POLO]:
3.58/1.77	
3.58/1.77	   POL(0) = 2
3.58/1.77	   POL(add(x_1, x_2)) = 2*x_1 + 2*x_2
3.58/1.77	   POL(and(x_1, x_2)) = 2 + x_1 + x_2
3.58/1.77	   POL(cons(x_1, x_2)) = 0
3.58/1.77	   POL(false) = 2
3.58/1.77	   POL(first(x_1, x_2)) = 2 + 2*x_1 + x_2
3.58/1.77	   POL(from(x_1)) = 0
3.58/1.77	   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.58/1.77	   POL(nil) = 2
3.58/1.77	   POL(s(x_1)) = 2
3.58/1.77	   POL(true) = 2
3.58/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.58/1.77	
3.58/1.77	   and(true, X) -> X
3.58/1.77	   and(false, Y) -> false
3.58/1.77	   if(true, X, Y) -> X
3.58/1.77	   if(false, X, Y) -> Y
3.58/1.77	   add(0, X) -> X
3.58/1.77	   add(s(X), Y) -> s(add(X, Y))
3.58/1.77	   first(0, X) -> nil
3.58/1.77	   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
3.58/1.77	
3.58/1.77	
3.58/1.77	
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(4)
3.58/1.77	Obligation:
3.58/1.77	Context-sensitive rewrite system:
3.58/1.77	The TRS R consists of the following rules:
3.58/1.77	
3.58/1.77	   from(X) -> cons(X, from(s(X)))
3.58/1.77	
3.58/1.77	The replacement map contains the following entries:
3.58/1.77	
3.58/1.77	s: empty set
3.58/1.77	cons: empty set
3.58/1.77	from: empty set
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(5) CSRRRRProof (EQUIVALENT)
3.58/1.77	The following CSR is given: Context-sensitive rewrite system:
3.58/1.77	The TRS R consists of the following rules:
3.58/1.77	
3.58/1.77	   from(X) -> cons(X, from(s(X)))
3.58/1.77	
3.58/1.77	The replacement map contains the following entries:
3.58/1.77	
3.58/1.77	s: empty set
3.58/1.77	cons: empty set
3.58/1.77	from: empty set
3.58/1.77	Used ordering:
3.58/1.77	Polynomial interpretation [POLO]:
3.58/1.77	
3.58/1.77	   POL(cons(x_1, x_2)) = 2*x_1
3.58/1.77	   POL(from(x_1)) = 1 + 2*x_1
3.58/1.77	   POL(s(x_1)) = 0
3.58/1.77	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.58/1.77	
3.58/1.77	   from(X) -> cons(X, from(s(X)))
3.58/1.77	
3.58/1.77	
3.58/1.77	
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(6)
3.58/1.77	Obligation:
3.58/1.77	Context-sensitive rewrite system:
3.58/1.77	R is empty.
3.58/1.77	
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(7) RisEmptyProof (EQUIVALENT)
3.58/1.77	The CSR R is empty. Hence, termination is trivially proven.
3.58/1.77	----------------------------------------
3.58/1.77	
3.58/1.77	(8)
3.58/1.77	YES
3.68/1.80	EOF